You are still reading your data one line at a time so your pattern matches are matching against one line at a time.

To see what is being read, you might experiment with the following program:

#!/usr/bin/perl -w
use strict;
use warnings;

while (<DATA>) {
    print "start of loop\n";
    print "\$_ = \"$_\"\n";
}

__DATA__
<a>a,
b</a>
<b>a,b</b>
<b>a,b
</b>
[download]

To make <DATA> read all your DATA instead of just one line, you can set $/ before your while loop, as follows. Note the use of a block ({ }) and local so that $/ isn't affected elsewhere in the program.

#!/usr/bin/perl -w
use strict;
use warnings;

{
    local $/;
    while (<DATA>) {
        print "start of loop\n";
        print "\$_ = \"$_\"\n";
    }
}

__DATA__
<a>a,
b</a>
<b>a,b</b>
<b>a,b
</b>
[download]

With $/ set to undef (which is what local $/ does), there is no need for the while loop - all the data is read on the first iteration. You can read all the data into a single variable as follows:

#!/usr/bin/perl -w
use strict;
use warnings;

my $var = do { local $/; <DATA> };
print "\$var = \"$var\"";

__DATA__
<a>a,
b</a>
<b>a,b</b>
<b>a,b
</b>
[download]

Then you can substitute against this variable as follows:

$var =~ s!<a>(.*?),(.*?)</a>!<a>$1A$2</a>!gs;
[download]

When you get that working, you might want to try with the following data:

__DATA__
This is to test.
<a>a,
b</a>
<b>a,b</b>
<b>a,b
</b>
<a>ab</a>
<c>a,b</c>
<a>a,b</a>
[download]